Improved bound for the dimension of posets of treewidth two

Seweryn, Michał T.

doi:10.1016/j.disc.2019.111605

Cited by 4 publications

(2 citation statements)

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“…The first major result linking dimension with structural graph theory is due to Joret, Micek, Milans, Trotter, Walczak and Wang [7], who showed that for each pair (t, h) of positive integers, there is a least positive integer d(t, h) so that if P is a poset of height h and the tree-width 1 of the cover graph of P is t, then dim Recently, Seweryn [13] has given the following substantive improvement:…”

Section: Structural Graph Theorymentioning

confidence: 99%

Local Dimension is Unbounded for Planar Posets

Bosek¹,

Grytczuk²,

Trotter³

2020

Electron. J. Combin.

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In 1981, Kelly showed that planar posets can have arbitrarily large dimension. However, the posets in Kelly's example have bounded Boolean dimension and bounded local dimension, leading naturally to the questions as to whether either Boolean dimension or local dimension is bounded for the class of planar posets. The question for Boolean dimension was first posed by Nešetřil and Pudlák in 1989 and remains unanswered today. The concept of local dimension is quite new, introduced in 2016 by Ueckerdt. Since that time, researchers have obtained many interesting results concerning Boolean dimension and local dimension, contrasting these parameters with the classic Dushnik-Miller concept of dimension, and establishing links between both parameters and structural graph theory, path-width and tree-width in particular. Here we show that local dimension is not bounded on the class of planar posets. Our proof also shows that the local dimension of a poset is not bounded in terms of the maximum local dimension of its blocks, and it provides an alternative proof of the fact that the local dimension of a poset cannot be bounded in terms of the tree-width of its cover graph, independent of its height.

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Section: Structural Graph Theorymentioning

confidence: 99%

Local Dimension is Unbounded for Planar Posets

Bosek¹,

Grytczuk²,

Trotter³

2020

Electron. J. Combin.

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“…Generally, in the past 15 years alone, more than 25 published research papers have explored connections between dimension and graph theoretic properties of cover graphs and order diagrams (we list some of them: outerplanarity [3,5], planarity [18,13,1,10], cut-vertices structure [22], treewidth [17,9], exclusion of some structures [25,15,6,7], or even more complex structural parameters [11,8]). For our considerations, it is worth mentioning a result in [9].…”

Section: Introductionmentioning

confidence: 99%

Forcing the wheel

Jędrzej

Trotter

2023

Preprint

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Over the past 10 years, there has been considerable interest in exploring questions connecting dimension for posets with graph theoretic properties of their cover graphs and order diagrams, especially with the concepts of planarity and treewidth. Joret and Micek conjectured that if P is a poset with a planar cover graph, then the dimension of P is bounded in terms of the number of minimal elements of P and the treewidth of the cover graph of P. We settle this conjecture in the affirmative. To this end, we strengthen a recent breakthrough result by Blake, Micek, and Trotter, who proved that for each poset P admitting a planar cover graph and a unique minimal element we have dim(P) <= 2 se(P) + 2, where dim(P) stands for the dimension of P and se(P) stands for the maximum order of a standard example contained in P. We prove that dim(P) <= 2 wheel(P) + 2, where wheel(P) stands for the maximum order of a wheel contained in P.

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